What is the general solution for cos(x-pi/3)+cos(x+pi/3)=1 ?

The general solution for cos(x-pi/3)+cos(x+pi/3)=1 is x=2pin

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cos(x-pi/3)+cos(x+pi/3)=1

Solution

cos (x - \frac{π}{3}) + cos (x + \frac{π}{3}) = 1

Solution

x = 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (x - \frac{π}{3}) + cos (x + \frac{π}{3}) = 1

Rewrite using trig identities

cos (x - \frac{π}{3}) + cos (x + \frac{π}{3})

Use the Sum to Product identity:

cos (s) + cos (t) = 2 cos (\frac{s + t}{2}) cos (\frac{s - t}{2})

= 2 cos (\frac{x - \frac{π}{3} + x + \frac{π}{3}}{2}) cos (\frac{x - \frac{π}{3} - ( x + \frac{π}{3} )}{2})

Simplify

2 cos (\frac{x - \frac{π}{3} + x + \frac{π}{3}}{2}) cos (\frac{x - \frac{π}{3} - ( x + \frac{π}{3} )}{2}) : cos (x)

2 cos (\frac{x - \frac{π}{3} + x + \frac{π}{3}}{2}) cos (\frac{x - \frac{π}{3} - ( x + \frac{π}{3} )}{2})

\frac{x - \frac{π}{3} + x + \frac{π}{3}}{2} = x

\frac{x - \frac{π}{3} + x + \frac{π}{3}}{2}

x - \frac{π}{3} + x + \frac{π}{3} = 2 x

x - \frac{π}{3} + x + \frac{π}{3}

Group like terms

= x + x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

x + x = 2 x

= 2 x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

= 2 cos (x) cos (\frac{x - ( x + \frac{π}{3} ) - \frac{π}{3}}{2})

\frac{x - \frac{π}{3} - ( x + \frac{π}{3} )}{2} = - \frac{π}{3}

\frac{x - \frac{π}{3} - ( x + \frac{π}{3} )}{2}

Join

x - \frac{π}{3} - (x + \frac{π}{3}) : - \frac{2 π}{3}

x - \frac{π}{3} - (x + \frac{π}{3})

Convert element to fraction:

x = \frac{x 3}{3}, (x + \frac{π}{3}) = \frac{( x + \frac{π}{3} ) 3}{3}

= \frac{x \cdot 3}{3} - \frac{π}{3} - \frac{( x + \frac{π}{3} ) \cdot 3}{3}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{x \cdot 3 - π - ( x + \frac{π}{3} ) \cdot 3}{3}

Expand

x \cdot 3 - π - (x + \frac{π}{3}) \cdot 3 : - 2 π

x \cdot 3 - π - (x + \frac{π}{3}) \cdot 3

= 3 x - π - 3 (x + \frac{π}{3})

Expand

- 3 (x + \frac{π}{3}) : - 3 x - π

- 3 (x + \frac{π}{3})

Apply the distributive law:

a (b + c) = ab + a c

a = - 3, b = x, c = \frac{π}{3}

= - 3 x + (- 3) \frac{π}{3}

Apply minus-plus rules

+ (- a) = - a

= - 3 x - 3 \cdot \frac{π}{3}

3 \cdot \frac{π}{3} = π

3 \cdot \frac{π}{3}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 3}{3}

Cancel the common factor:

3

= π

= - 3 x - π

= x \cdot 3 - π - 3 x - π

Simplify

x \cdot 3 - π - 3 x - π : - 2 π

x \cdot 3 - π - 3 x - π

Group like terms

= 3 x - 3 x - π - π

Add similar elements:

3 x - 3 x = 0

= - π - π

Add similar elements:

- π - π = - 2 π

= - 2 π

= - 2 π

= \frac{- 2 π}{3}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2 π}{3}

= \frac{- \frac{2 π}{3}}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{2 π}{3}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

\frac{\frac{2 π}{3}}{2} = \frac{2 π}{3 \cdot 2}

= - \frac{2 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= - \frac{2 π}{6}

Cancel the common factor:

2

= - \frac{π}{3}

= 2 cos (- \frac{π}{3}) cos (x)

cos (- \frac{π}{3}) = \frac{1}{2}

cos (- \frac{π}{3})

Use the following property:

cos (- x) = cos (x)

cos (- \frac{π}{3}) = cos (\frac{π}{3})

= cos (\frac{π}{3})

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

= \frac{1}{2}

= 2 \cdot \frac{1}{2} cos (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2} cos (x)

Cancel the common factor:

2

= cos (x) \cdot 1

Multiply:

cos (x) \cdot 1 = cos (x)

= cos (x)

= cos (x)

cos (x) = 1

General solutions for

cos (x) = 1

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x-pi/3)+cos(x+pi/3)=1 ?
The general solution for cos(x-pi/3)+cos(x+pi/3)=1 is x=2pin